Theorem caovordg 6841

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypothesis

Ref	Expression
caovordg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )

Assertion

Ref	Expression
caovordg	|- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ph, x, y, z   x, F, y, z   x, R, y, z   x, S, y, z

Proof of Theorem caovordg

Step	Hyp	Ref	Expression
1		caovordg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
2	1	ralrimivvva 2972	. 2 |- ( ph -> A. x e. S A. y e. S A. z e. S ( x R y <-> ( z F x ) R ( z F y ) ) )
3		breq1 4656	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 6658	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 4663	. . . 4 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
6	3, 5	bibi12d 335	. . 3 |- ( x = A -> ( ( x R y <-> ( z F x ) R ( z F y ) ) <-> ( A R y <-> ( z F A ) R ( z F y ) ) ) )
7		breq2 4657	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 6658	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 4665	. . . 4 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
10	7, 9	bibi12d 335	. . 3 |- ( y = B -> ( ( A R y <-> ( z F A ) R ( z F y ) ) <-> ( A R B <-> ( z F A ) R ( z F B ) ) ) )
11		oveq1 6657	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 6657	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 4666	. . . 4 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
14	13	bibi2d 332	. . 3 |- ( z = C -> ( ( A R B <-> ( z F A ) R ( z F B ) ) <-> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
15	6, 10, 14	rspc3v 3325	. 2 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( x R y <-> ( z F x ) R ( z F y ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) ) )
16	2, 15	mpan9 486	1 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B <-> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovordd  6842